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\begin{document}

\title {Anisotropic Wave Propagation}

\author {Anton Kodochygov}

\begin{frame}
\titlepage
\end{frame}
\begin{frame}

\frametitle{Isotropic Elastic Modulus Matrix}

\[ C =  \left( \begin{array}{cccccc}
C_{33} & C_{12} & C_{12} & 0 & 0 & 0 \\
C_{12} & C_{33} & C_{12} & 0 & 0 & 0 \\
C_{12} & C_{12} & C_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & C_{55} & 0 & 0 \\
0 & 0 & 0 & 0 & C_{55} & 0 \\
0 & 0 & 0 & 0 & 0 & C_{55} \end{array} \right)\] 

\begin{eqnarray}
C_{12} &=& C_{33} - 2 C_{55} \nonumber
\end{eqnarray}

\end{frame}

\begin{frame}
\frametitle{VTI Elastic Modulus Matrix}

\[ C =  \left( \begin{array}{cccccc}
C_{11} & C_{11}-2C_{66} & C_{13} & 0 & 0 & 0 \\
C_{11}-2C_{66} & C_{11} & C_{13} & 0 & 0 & 0 \\
C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & C_{44} & 0 & 0 \\
0 & 0 & 0 & 0 & C_{44} & 0 \\
0 & 0 & 0 & 0 & 0 & C_{66} \end{array} \right)\] 

\end{frame}

\begin{frame}
\frametitle{TTI Elastic Modulus Matrix}

$C_{TTI} = M C_{VTI} M^T$

M - Bond rotation matrix

\end{frame}

\begin{frame}
\frametitle{Exact Phase Velocities}
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\begin{eqnarray}
V_{qP}(\xi) &=& \sqrt{\frac{C_{11}sin^2(\xi)+C_{33}cos^2(\xi)+C_{44}+\sqrt{M(\xi)}}{2\rho}} \nonumber \\
V_{qS}(\xi) &=& \sqrt{\frac{C_{11}sin^2(\xi)+C_{33}cos^2(\xi)+C_{44}-\sqrt{M(\xi)}}{2\rho}} \nonumber \\
V_{S}(\xi) &=& \sqrt{\frac{C_{66}sin^2(\xi)+C_{44}cos^2(\xi)}{\rho}} \nonumber \\
M(\xi) &=& \left[(C_{11}-C_{44})sin^2(\xi)-(C_{33}-C_{44})cos^2(\xi)\right]^2 + (C_{13}+C_{44})^2sin^2(2\xi) \nonumber
\end{eqnarray}

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\begin{frame}
\frametitle{Thomson Approximation - Weak Anisotropy $\delta,\gamma,\epsilon << 1$}
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\begin{eqnarray}
V_{qP}(\xi) &\approx& V_{P0} (1 + \delta sin^2(\xi)cos^2(\xi) + \epsilon sin^4(\xi))\nonumber \\
V_{qS}(\xi) &\approx& V_{S0} \left[ 1 + \left(\frac{V_{P0}}{V_{S0}}\right)^2 (\epsilon-\delta)sin^2(\xi)cos^2(\xi) \right]\nonumber \\
V_{S}(\xi)  &\approx& V_{S0} (1 + \gamma sin^2(\xi))\nonumber
\end{eqnarray}

\end{frame}

\begin{frame}
\frametitle{Thomson Parameers from Elastic Moduli}
\begin{eqnarray}
\epsilon &=& \frac{C_{11}-C_{33}}{2C_{33}} \nonumber \\
\delta &=& \frac{(C_{13}+C_{44})^2-(C_{33}-C_{44})^2}{2C_{33}(C_{33}-C_{44})} \nonumber \\
\gamma &=& \frac{C_{66}-C_{44}}{2C_{44}} \nonumber \\
V_{P0} &=& \sqrt{\frac{C_{33}}{\rho}} \nonumber \\
V_{S0} &=& \sqrt{\frac{C_{44}}{\rho}} \nonumber 
\end{eqnarray}
\end{frame}

\begin{frame}
\frametitle{Elastic Moduli from Thomson Parameters}
\begin{eqnarray}
C_{33} &=& V_{P0}^2 \rho \nonumber \\
C_{44} &=& V_{S0}^2 \rho \nonumber \\
C_{11} &=& 2 \epsilon C_{33} + C_{33} \nonumber \\
C_{66} &=& 2 \gamma C_{44} + C_{44} \nonumber \\
C_{13} &=& \sqrt{2 \delta C_{33} (C_{33} - C_{44}) + (C_{33} - C_{44})^2} - C_{44} \nonumber 
\end{eqnarray}
\end{frame}

\begin{frame}
\frametitle{Isotropic Wave Propagation}
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\begin{eqnarray}
\rho \frac{\partial^2 U}{\partial t^2} = \nabla^2 U + S \nonumber \\
\rho_{i,j} \left( \frac{ U^{n+1}_{i,j} - 2 U^n_{i,j} + U^{n-1}_{i,j} }{ \Delta t^2 } \right) = \left( \frac{ U^n_{i+1,j} - 2 U^n_{i,j} + U^n_{i-1,j} }{ \Delta x^2 } \right) + \left( \frac{ U^n_{i,j+1} - 2 U^n_{i,j} + U^n_{i,j-1} }{ \Delta z^2 } \right) + S^n_{i,j} \nonumber \\
U^{n+1}_{i,j} = 2 U^n_{i,j} - U^{n-1}_{i,j} + \frac{\Delta t^2}{\Delta x^2 \rho_{i,j} }\left( U^n_{i+1,j} - 2 U^n_{i,j} + U^n_{i-1,j} \right) + \frac{\Delta t^2}{\Delta z^2 \rho_{i,j} }\left( U^n_{i,j+1} - 2 U^n_{i,j} + U^n_{i,j-1} \right) + \frac{\Delta t^2}{\rho_{i,j}}S^n_{i,j} \nonumber
\end{eqnarray}
\end{frame}

\begin{frame}
\frametitle{Anisotropic Wave Propagation}
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\begin{eqnarray}
\rho \frac{\partial^2 U}{\partial t^2} = \frac{\partial}{\partial x} \left( A \frac{\partial}{\partial x} + C \frac{\partial}{\partial z} \right) U + \frac{\partial}{\partial z} \left( G \frac{\partial}{\partial x} + Q \frac{\partial}{\partial z} \right) U + S \nonumber \\
\rho_{i,j} \left( \frac{ U^{n+1}_{i,j} - 2 U^n_{i,j} + U^{n-1}_{i,j} }{ \Delta t^2 } \right) = \left( \frac{ A_{i+1/2,j}(U^n_{i+1,j} - U^n_{i,j}) - A_{i-1/2,j}( U^n_{i,j}- U^n_{i-1,j}) }{ \Delta x^2 } \right) \nonumber \\
    + \left( \frac{ Q_{i,j+1/2} ( U^n_{i,j+1} - U^n_{i,j} ) - Q_{i,j-1/2} ( U^n_{i,j} - U^n_{i,j-1} ) }{ \Delta z^2 } \right) \nonumber \\
    + \left( \frac{ C_{i+1,j} ( U^n_{i+1,j+1} - U^n_{i+1,j-1} ) - C_{i-1,j} ( U^n_{i-1,j+1} - U^n_{i-1,j-1} ) }{ 4 \Delta x \Delta z } \right) \nonumber \\
    + \left( \frac{ G_{i,j+1} ( U^n_{i+1,j+1} - U^n_{i-1,j+1} ) - G_{i,j-1} ( U^n_{i+1,j-1} - U^n_{i-1,j-1} ) }{ 4 \Delta x \Delta z } \right) \nonumber \\
    + S^n_{i,j} \nonumber \\
    A_{i+1/2,j} = \frac{1}{2} (A_{i+1,j} + A_{i,j}) \nonumber \\
    Q_{i,j+1/2} = \frac{1}{2} (Q_{i,j+1} + Q_{i,j}) \nonumber 
\end{eqnarray}
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\end{document}

